Steady motion of 80-nm-size skyrmions in a 100-nm-wide track

The current-driven movement of magnetic skyrmions along a nanostripe is essential for the advancement and functionality of a new category of spintronic devices resembling racetracks. Despite extensive research into skyrmion dynamics, experimental verification of current-induced motion of ultra-small skyrmions within an ultrathin nanostripe is still pending. Here, we unveil the motion of individual 80 nm-size skyrmions in an FeGe track with an ultrathin width of 100 nm. The skyrmions can move steadily along the track over a broad range of current densities by using controlled pulse durations of as low as 2 ns. The potential landscape, arising from the magnetic edge twists in such a geometrically confined system, introduces skyrmion inertia and ensures efficient motion with a vanishing skyrmion Hall angle. Our results showcase the steady motion of skyrmions in an ultrathin track, offering a practical pathway for implementing skyrmion-based spintronic devices.


Supplementary Note I -Micromagnetic framework of skyrmion dynamics
In chiral magnets such as FeGe, the current flow will result in torques, which rotate the local magnetization.This process can be described by the Landau-Lifshitz-Gilbert (LLG) equation, including spin-transfer torques (Zhang-Li model), according to the expression 1 !" where $ is a unit vector of the magnetization, g is the gyromagnetic ratio, & $%% is the total effective field and a is the Gilbert damping.The third and fourth terms on the right side of the equation are the spin-transfer torques.The parameter u is defined as where 1 is the current density, 2 is the Landé factor, µ , is the Bohr magneton, 4 (> 0) is the electron charge, P is the polarization rate of the current, 5 .is the saturation magnetization and β is the nonadiabatic spin-transfer parameter 2 .

Supplementary Note II -Analytical descriptions for skyrmion motion and skyrmion inertia on the track
For a one-dimensional nanotrack, the twist at the edges imposes an extra potential for the skyrmion.As a reasonable approximation, we consider a quadratic potential , where Y is the y-component of the skyrmion's center.Ignoring the influence of disorder, we obtain where τ = is the characteristic time, and ] = ([( 7), W( 7)) describe the skyrmion position.For a DC current, the skyrmion velocity will increase to the enhanced steady speed in Eq. ( 6) and the y-component of the skyrmion center will saturate eventually.
It is worth mentioning that the stable moving region for the skyrmion in a narrow track is limited, which requires that |W(7)| < W K /2, where W K is the width of the stable moving region and a rough estimate gives W K ~40 bc for our 100-nm-wide track.It can be seen from Eq. ( 7) that either a low current density u or a short current pulse is sufficient for stable skyrmion motion.Therefore, the critical pulse width d K , beyond which the skyrmion will be eliminated can be obtained: The critical pulse width is direct related to the current density u.For a sufficiently low , |*'/| , the skyrmion can move steadily even for a DC current.
After switching off the current pulse, the y-component of the skyrmion will decrease to zero again due to the edge repulsion and its x-coordinate will continue to increase in this process: where W L = W(#) and w is the pulse width.For the situation when % > (, T 9 is always positive for both C = ±1.The displacement due to inertia after switching off the pulse is given by the expression Interestingly, if the velocity under the current pulse is defined as T̅ = Meanwhile, the y-component of the skyrmion takes a relatively small value of ~ 5 nm under the current pulse and then decreases to zero after switching off the pulse due to the edge repulsion, as shown in Supplementary Fig. 9c.The critical pulse width d K calculated using Eq. ( 11) is shown in Supplementary Fig. 9d, beyond which the skyrmion will be erased at the edge.These results are consistent with the main text's micromagnetic simulations in Fig. 3.

Supplementary Note III -Estimate of information-density in the 100-nm-wide nanotrack
It is possible to give a rough estimate of the areal density disregarding the influence of pinning effects.Based on the skyrmion-skyrmion interactions described before 7 , the safe skyrmion-skyrmion spacing would likely need to exceed 200 nm in FeGe.This roughly corresponds to a maximum information-density of approximately 4Gb/inch 2 in a 100-nm-wide nanotrack.The density can be further increased by selecting materials with smaller skyrmion sizes.The minimum spacing is closely related to the size of skyrmion.For materials with skyrmion sizes around several nanometers, the density can be increased significantly.The position of skyrmion can also be controlled by external means such as artificial pinning sites (e.g., patterning notches along the edges).
Therefore, the minimum skyrmion-skyrmion spacing to avoid the non-synchronous motion may be further reduced.similar to the method used for single or two skyrmions above.However, in this case, the magnetic field is maintained at a relatively higher value of ~150 mT.This higher magnetic field is chosen to prevent the creation of too many skyrmions by the thermal effects.By applying a high current density of u = 12.75 × 10 10 v • c B) , skyrmion chains can be directly generated with a certain probability as shown in Supplementary Fig. 4b with a skyrmion number of 4.However, it should be noted that we cannot control the exact number of skyrmions in the chains in a deterministic manner.This method allows for the creation of varying numbers of skyrmions, but the specific quantity in each chain is subject to variability inherent to the process.

Supplementary
For a magnetic skyrmion without distortion, D 34 = L 34 D and D is typically close to unity.The pinning force < 567 originates from disorder in the material and is given by < 3 567 = ∫ #$ • [H 3 $ × ($ × & 567 )] dJ dK .The environmental force can be defined as < = −,U and N = N(O) is the phenomenological environmental potential representing the pushing force imposed by spin twisting at the edges.Within the framework of micromagnetic theory, the phenomenological potential is directly connected to the total micromagnetic energy E, i.e., N = (#/µ 8 5 .P)Q, where L is the thickness of the sample.
we find that T̅ = C A R, which is the same as the skyrmion velocity under a DC current.Supplementary Fig.9ashows the potential obtained using the micromagnetic simulations for a 100-nm-wide nanotrack for an external field of k = 100 mT.The potential is fitted using a quadratic function N(W) = = ) XW ) , with a fitted value of X = 25.6 ns B= .For typical parameters for skyrmions G = 4B, D = 4B × 1.2, ( = 0.0167, the characteristic time is established to be τ = 24.5 ns.Supplementary Fig.9bshows the skyrmion displacement under a current pulse based on Eqs.(9-15), indicating that the skyrmion continues to move under inertia after switching off the current pulse.

Fig. 1 | 2 |
In-situ electrical Lorentz TEM experimental setup of FeGe micro-device.a, Optical image of the end of a Gatan TEM specimen holder, which was designed for in situ electrical biasing and cooling experiments.The four electrical ports were designed by Gatan to be connected to an external voltage source.In order to build the circuit between the ports and the sample, a customized electrical chip with four Au electrodes was self-designed, as shown in b.The Au pads were connected to the ports using Cu wires, which were manually fixed using silver colloidal paste (only two ports were used in our experiments).The electrical TEM FeGe micro-device was fabricated using a FIB workstation, as such a chip makes the fabrication process compatible with the conventional FIB lift-out method.c, Low magnification SEM image of the FeGe electrical device used for in situ Lorentz TEM experiments.The FeGe nanostripe was covered by amorphous carbon on the upper and lower layers.The left and right edges were connected using two Pt electrodes, which were in turn connected to the source of electrical current pulses through the Au electrodes.Procedure for the fabrication of a 100-nm-wide FeGe nanostripe for in situ Lorentz TEM using a FIB-SEM dual-beam system.a, An FeGe membrane was carved on the surface of an FeGe single crystal after milling two trenches on each side of a carbon protection layer.b, The height of the central membrane was milled down to ~1.2 μm by cutting off the lower part.c, Top view of the central membrane after thinning to a thickness of 100 nm, which was the desired width of the nanostripe.The surfaces of the membrane were polished using a small beam current to reduce the amorphous/damaged layer.d, The 100-nm-thick nanostripe was protected by carbon layers deposited on each surface.e, The carbon-encapsulated membrane was rotated by 90° and fixed onto a customized electrical chip.The left and right ends of the nanostripe were connected to Au electrodes on the chip by fabricating Pt nanosticks.f, The FeGe nanostripe was thinned to ~150 nm and the surfaces were polished for TEM observation.

Fig. 7 |
Skyrmion motion under varied magnetic fields.a, The lower (motion) and upper (annihilation) critical current density with a pulse duration of 5 ns as a function of magnetic field.The lower critical current density corresponds to the threshold at which the skyrmion begins to move, while the upper critical current density is the point at which the skyrmion is annihilated upon application of the current pulse.b, Skyrmion velocities plotted as a function of current density under different magnetic fields.The pulse duration is 5 ns.